1964 AHSME Problems/Problem 9

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Problem

A jobber buys an article at $24$ less $12\frac{1}{2}\%$. He then wishes to sell the article at a gain of $33\frac{1}{3}\%$ of his cost after allowing a $20\%$ discount on his marked price. At what price, in dollars, should the article be marked?

$\textbf{(A)}\ 25.20 \qquad \textbf{(B)}\ 30.00 \qquad \textbf{(C)}\ 33.60 \qquad \textbf{(D)}\ 40.00 \qquad \textbf{(E)}\ \text{none of these}$

Solution

The item is bought for $24$ minus $12\frac{1}{2}\%$ of $24$. Since $12\frac{1}{2}\% = \frac{1}{8}$, the item was purchased for $24 - 24 \cdot \frac{1}{8}$, or $21$.

He wants to make $33\frac{1}{3}\%$ of the purchase price, or $21 \cdot \frac{1}{3} = 7$. This means the price must actually sell for $21 + 7 = 28$.

If the article were marked for $x$ dollars, then after the $20\%$ discount, it would sell for $\frac{4}{5}x$. Thus, we have $\frac{4}{5}x = 28$, which leads to $x = 28 \cdot \frac{5}{4}$, or $35$ dollars. The correct answer is $\boxed{\textbf{(E)}}$